Old English felan "to touch or have a sensory experience of; perceive, sense (something)," in late Old English "have a mental perception," from Proto-Germanic *foljanan (source also of Old Saxon gifolian, Old Frisian fela, Dutch voelen, Old High German vuolen, German fühlen "to feel," Old Norse falma "to grope"), which is of uncertain origin, perhaps from PIE root *pal- "to touch, feel, shake, strike softly" (source also of Greek psallein "to pluck (the harp)," Latin palpare "to touch softly, stroke," palpitare "to move quickly"), perhaps ultimately imitative.
In Germanic languages, the specific word for "perceive by sense of touch" has tended to evolve to apply to the emotions. The connecting notion might be "perceive through senses which are not referred to any special organ." Sense of "be conscious of a tactile sensation, sense pain, pleasure, illness, etc.; have an emotional experience or reaction," developed by c. 1200, also "have an opinion or conviction;" that of "to react with sympathy or compassion" is from mid-14c. Meaning "to try by touch" is from early 14c. From late 14c. as "know (something) beforehand, to have foreknowledge of." To feel like "want to" attested from 1829.